Problem 9 - Algebra and Trigonometry

Question

Solve the equation by factoring. \(6 x^{2}+5 x=4\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = -\frac{4}{3}\).

Step 1: Move All Terms to One Side

Start by moving the constant term to the left side of the equation to set it equal to zero, as this sets up the equation for factoring:\[ 6x^2 + 5x - 4 = 0 \] This equation is in the standard quadratic form \( ax^2 + bx + c = 0 \).

Step 2: Identify Coefficients

Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation:\[ a = 6, \quad b = 5, \quad c = -4 \] These will be used to factor the polynomial.

Step 3: Factor by Grouping

Since the quadratic trinomial is in the form \( ax^2 + bx + c \), we need to find two numbers that multiply to \( a \times c = 6 imes (-4) = -24 \) and add to \( b = 5 \).The numbers 8 and -3 meet these conditions because \( 8 imes (-3) = -24 \) and \( 8 + (-3) = 5 \).Rewrite the middle term using these numbers: \[ 6x^2 + 8x - 3x - 4 = 0 \]

Step 4: Group and Factor

Group the first two terms and the last two terms: \[ (6x^2 + 8x) + (-3x - 4) = 0 \]Factor out the greatest common factor from each group:\[ 2x(3x + 4) - 1(3x + 4) = 0 \]

Step 5: Factor Out the Common Binomial

Notice that \((3x + 4)\) is a common factor:\[ (2x - 1)(3x + 4) = 0 \]

Step 6: Solve Each Factor

Set each factor equal to zero and solve for \(x\):For \(2x - 1 = 0\):\[ 2x = 1 \]\[ x = \frac{1}{2} \]For \(3x + 4 = 0\):\[ 3x = -4 \]\[ x = -\frac{4}{3} \]

Key Concepts

quadratic formula

The quadratic formula is a powerful tool used to solve equations of the form \( ax^2 + bx + c = 0 \). It provides a way to find the values of \( x \) without needing to factor the equation manually. The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how it works:

• \( a \), \( b \), and \( c \) are the coefficients from the standard form of the quadratic equation.
• The expression under the square root sign, \( b^2 - 4ac \), is called the discriminant. It determines the nature of the roots of the equation. A positive discriminant means two distinct real roots, zero means one real root (or a repeated root), and negative means two complex roots.
• By substituting the coefficients into the quadratic formula, you can calculate the solutions for \( x \).

Using the quadratic formula can be especially helpful when the equation is too complex for easy factoring or when coefficients are not integers.

polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, arranged in terms of non-negative integer powers of the variables. A general polynomial can be expressed as:\[ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]Where:

• \( n \) is a non-negative integer (the degree of the polynomial).
• \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients, with \( a_n eq 0 \).
• Each term consists of a coefficient (like \( a_k \)) multiplied by a variable raised to a power (like \( x^k \)).

Quadratics are a specific type of polynomial where the highest power of the variable is 2, such as \( ax^2 + bx + c \). Understanding polynomials is crucial because they can be used to represent complex relationships and can be solved or simplified using methods such as factoring or using the quadratic formula. Recognizing the degree of the polynomial helps determine the number of solutions or roots it may have.

solving equations

Solving equations involves finding the values of the variable that make the equation true. For quadratic equations, the goal is to find values of \( x \) such that the equation equals zero. Several methods can be used to solve quadratic equations:

Factoring

Factoring involves rewriting the equation as a product of its factors, which can then be set to zero and solved individually. This is useful when the equation is factorable with whole number coefficients.

Using the Quadratic Formula

This method is versatile and can solve any quadratic equation, especially when factoring is complex or impossible.

Completing the Square

This involves rearranging the equation to form a perfect square trinomial, allowing for easy solving of the variable.Each method has its uses depending on the form and complexity of the quadratic equation. The best approach often depends on the specific equation at hand and whether coefficients are integers or fractions. Understanding these methods enhances problem-solving skills and boosts confidence in tackling varied mathematical challenges.